Abstract: In the numerical simulation of many practical problems of fluid mechanics, the finite volume method is a very important and popular numerical method due to the local conservation and the capability of discretizing domains with complex geometry. A second-order mixed finite volume method is proposed in this paper to solve the Navier-Stokes equations. Specifically, on the triangular meshes, the trial function space for velocity of the Navier-Stokes equations is taken as the hierarchical quadratic conforming finite element space, and the corresponding test function space is composed of the piecewise constant functions and the piecewise quadratic polynomial functions. Both the trial function space and the test function space for pressure are chosen as the piecewise linear finite element space. The nonlinear terms in the Navier-Stokes equations are directly discretized on the control volumes of the finite volume method. Under the standard assumption that the viscosity parameter satisfies certain conditions, the stability of the proposed mixed finite volume method is proved and the optimal order error estimates for velocity and pressure of the Navier-Stokes equations are obtained, which are consistent with that of the corresponding finite element method.  The numerical examples are presented to verify the correctness of the theoretical results  and the effectiveness of the proposed numerical method.

Key words: finite volume method, Navier-Stokes equations, stability, error estimates